Previous |  Up |  Next

Article

Keywords:
Morrey space; Schrödinger operator; Riesz transform; fractional integral; Calderón-Zygmund estimate
Summary:
We first introduce new weighted Morrey spaces related to certain non-negative potentials satisfying the reverse Hölder inequality. Then we establish the weighted strong-type and weak-type estimates for the Riesz transforms and fractional integrals associated to Schrödinger operators. As an application, we prove the Calderón-Zygmund estimates for solutions to Schrödinger equation on these new spaces. Our results cover a number of known results.
References:
[1] Adams, D. R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53 (2004), 1629-1663. DOI 10.1512/iumj.2004.53.2470 | MR 2106339 | Zbl 1100.31009
[2] Alvarez, J., Bagby, R. J., Kurtz, D. S., Pérez, C.: Weighted estimates for commutators of linear operators. Stud. Math. 104 (1993), 195-209. DOI 10.4064/sm-104-2-195-209 | MR 1211818 | Zbl 0809.42006
[3] Bongioanni, B., Harboure, E., Salinas, O.: Riesz transforms related to Schrödinger operators acting on BMO type spaces. J. Math. Anal. Appl. 357 (2009), 115-131. DOI 10.1016/j.jmaa.2009.03.048 | MR 2526811 | Zbl 1180.42013
[4] Bongioanni, B., Harboure, E., Salinas, O.: Classes of weights related to Schrödinger operators. J. Math. Anal. Appl. 373 (2011), 563-579. DOI 10.1016/j.jmaa.2010.08.008 | MR 2720705 | Zbl 1203.42029
[5] Bui, T. A.: The weighted norm inequalities for Riesz transforms of magnetic Schrödinger operators. Differ. Integral Equ. 23 (2010), 811-826. MR 2675584 | Zbl 1240.42034
[6] Bui, T. A.: Weighted estimates for commutators of some singular integrals related to Schrödinger operators. Bull. Sci. Math. 138 (2014), 270-292. DOI 10.1016/j.bulsci.2013.06.007 | MR 3175023 | Zbl 1284.42068
[7] Coifman, R. R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51 (1974), 241-250. DOI 10.4064/sm-51-3-241-250 | MR 0358205 | Zbl 0291.44007
[8] Coulhon, T., Duong, X. T.: Riesz transforms for $ 1\leq p\leq 2$. Trans. Am. Math. Soc. 351 (1999), 1151-1169. DOI 10.1090/S0002-9947-99-02090-5 | MR 1458299 | Zbl 0973.58018
[9] Cruz-Uribe, D., Fiorenza, A.: Weighted endpoint estimates for commutators of fractional integrals. Czech. Math. J. 57 (2007), 153-160. DOI 10.1007/s10587-007-0051-y | MR 2309956 | Zbl 1174.42013
[10] Duong, X. T., Xiao, J., Yan, L.: Old and new Morrey spaces with heat kernel bounds. J. Fourier Anal. Appl. 13 (2007), 87-111. DOI 10.1007/s00041-006-6057-2 | MR 2296729 | Zbl 1133.42017
[11] Dziubański, J., Garrigós, G., Martínez, T., Torrea, J. L., Zienkiewicz, J.: BMO spaces related to Schrödinger operators with potentials satisfying reverse Hölder inequality. Mat. Z. 249 (2005), 329-356. DOI 10.1007/s00209-004-0701-9 | MR 2115447 | Zbl 1136.35018
[12] Dziubański, J., Zienkiewicz, J.: $H^{p}$ spaces for Schrödinger operators. Fourier Analysis and Related Topics W. Żelazko Banach Center Publications 56, Polish Academy of Sciences, Institute of Mathematics, Warsaw (2002), 45-53. DOI 10.4064/bc56-0-4 | MR 1971563 | Zbl 1039.42018
[13] Feuto, J., Fofana, I., Koua, K.: Spaces of functions with integrable fractional mean on locally compact groups. Afr. Mat., Sér. III French 15 (2003), 73-91. MR 2031873 | Zbl 1047.43004
[14] Feuto, J., Fofana, I., Koua, K.: Integrable fractional mean functions on spaces of homogeneous type. Afr. Diaspora J. Math. 9 (2010), 8-30. MR 2516238 | Zbl 1239.43002
[15] Fofana, I.: Study of a class of function spaces containing Lorentz spaces. French Afr. Mat. (2) 1 (1988), 29-50. MR 1080380 | Zbl 1210.46022
[16] Guo, Z., Li, P., Peng, L.: $L^{p}$ boundedness of commutators of Riesz transforms associated to Schrödinger operator. J. Math. Anal. Appl. 341 (2008), 421-432. DOI 10.1016/j.jmaa.2007.05.024 | MR 2394095 | Zbl 1140.47035
[17] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415-426. DOI 10.1002/cpa.3160140317 | MR 0131498 | Zbl 0102.04302
[18] Johnson, R., Neugebauer, C. J.: Change of variable results for $A_{p}$- and reverse Hölder $RH_{r}$-classes. Trans. Am. Math. Soc. 328 (1991), 639-666. DOI 10.2307/2001798 | MR 1018575 | Zbl 0756.42015
[19] Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282 (2009), 219-231. DOI 10.1002/mana.200610733 | MR 2493512 | Zbl 1160.42008
[20] Ly, F. K.: Second order Riesz transforms associated to the Schrödinger operator for $p\leq 1$. J. Math. Anal. Appl. 410 (2014), 391-402. DOI 10.1016/j.jmaa.2013.08.049 | MR 3109848 | Zbl 1319.42020
[21] Morrey, C.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43 (1938), 126-166. DOI 10.2307/1989904 | MR 1501936 | Zbl 0018.40501
[22] Muckenhoupt, B., Wheeden, R. L.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192 (1974), 261-274. DOI 10.2307/1996833 | MR 0340523 | Zbl 0289.26010
[23] Peetre, J.: On the theory of $\mathcal L^{p,\lambda}$ spaces. J. Funct. Anal. 4 (1969), 71-87. DOI 10.1016/0022-1236(69)90022-6 | MR 0241965 | Zbl 0175.42602
[24] Rao, M. M., Ren, Z. D.: Theory of Orlicz Spaces. Pure and Applied Mathematics 146, Marcel Dekker, New York (1991). MR 1113700 | Zbl 0724.46032
[25] Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350 (2009), 56-72. DOI 10.1016/j.jmaa.2008.09.021 | MR 2476892 | Zbl 1155.42005
[26] Segovia, C., Torrea, J. L.: Weighted inequalities for commutators of fractional and singular integrals. Publ. Mat., Barc. 35 (1991), 209-235. DOI 10.5565/PUBLMAT_35191_09 | MR 1103616 | Zbl 0746.42012
[27] Shen, Z.: $L^{p}$ estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier 45 (1995), 513-546. DOI 10.5802/aif.1463 | MR 1343560 | Zbl 0818.35021
[28] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43, Princeton University Press, Princeton (1993). MR 1232192 | Zbl 0821.42001
[29] Tang, L.: Weighted norm inequalities for Schrödinger type operators. Forum Math. 27 (2015), 2491-2532. DOI 10.1515/forum-2013-0070 | MR 3365805 | Zbl 1319.42014
[30] Tang, L., Dong, J.: Boundedness for some Schrödinger type operators on Morrey spaces related to certain nonnegative potentials. J. Math. Anal. Appl. 355 (2009), 101-109. DOI 10.1016/j.jmaa.2009.01.043 | MR 2514454 | Zbl 1166.35321
[31] Wang, H.: Boundedness of fractional integral operators with rough kernels on weighted Morrey spaces. Acta Math. Sin., Chin. Ser. Chinese. English summary 56 (2013), 175-186. MR 3097397 | Zbl 1289.42057
[32] Xiao, J.: Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn. Partial Differ. Equ. 4 (2007), 227-245. DOI 10.4310/DPDE.2007.v4.n3.a2 | MR 2353632 | Zbl 1147.42008
[33] Zhang, P.: Weighted endpoint estimates for commutators of Marcinkiewicz integrals. Acta Math. Sin., Engl. Ser. 26 (2010), 1709-1722. DOI 10.1007/s10114-010-8562-0 | MR 2672812 | Zbl 1202.42043
[34] Zhong, J.: Harmonic analysis for some Schrödinger type operators. Ph.D. Thesis, Princeton University (1993). MR 2689454
Partner of
EuDML logo